Recently, there has been significant interest in the tensor product property for cohomological support varieties of Hopf algebras and tensor categories. We will describe a method for approaching the tensor product property by way of a noncommutative version of Balmer’s tensor triangular geometry in the general setting of a monoidal triangulated category. We prove related properties about the collections of thick one-sided and two-sided ideals of the category, and then are often able to use the universal properties of the Balmer support to obtain applications to cohomological supports. Examples arising from the representation theory of Hopf algebras will be discussed throughout. This is joint work with Daniel Nakano and Milen Yakimov.